Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain
type: Item
instance of: scholarly article
title: Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain
zbMATH Open document ID: 1058.35028
published in: Journal of Differential Equations
publication date: 2004-10-01T00:00:00Z
review text: Under consideration are the reaction-diffusion equations \[ (*)_\varepsilon\quad\quad u_t=\Delta u+f(x,u) \quad\text{in}\quad \Omega_\varepsilon \] accompanying with the Neumann boundary condition \(\partial u/\partial n=0\) in \(\partial\Omega_\varepsilon,\) where \(\Omega_\varepsilon\) (\(0\leq\varepsilon\leq 1\)) are bounded Lipschitz domains in \({\mathbb R}^N\) (\(N\geq 2\)). Under the usual conditions on the nonlinearity \(f(x,u)\) which ensure the existence of an attractor \({\mathcal A}_\varepsilon\) for each \((*)_\varepsilon\) with respect to the \(H^1\)-topology, the authors show the convergence of the attractors \({\mathcal A}_\varepsilon\) as \(\varepsilon\to 0\) to the attractor \({\mathcal A}_0\) for the unperturbed problem \((*)_0\) (i.e., the case \(\varepsilon=0\)) if, besides an easy condition on the domains, the following spectral condition is satisfied: Every equilibrium of the unperturbed problem \((*)_0\) is hyperbolic.
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 37L15
zbMATH DE Number: 2105464
zbMATH Keywords: reaction-diffusion equations
zbMATH Keywords: stabilty
zbMATH Keywords: Neumann boundary condition
zbMATH Keywords: Lower semi-continuity
zbMATH Keywords: Upper semi-continuity
author: Jose M. Arrieta
author: A. N. Carvalho
MaRDI profile type: Publication
full work available at URL: j.jde.2003.09.004
OpenAlex ID: W2090851101
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DOI: 10.1016/J.JDE.2003.09.004
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description: scientific article; zbMATH DE number 2105464